Methods and systems for analyzing images in convolutional neural networks

ABSTRACT

A method for analyzing images to generate a plurality of output features includes receiving input features of the image and performing Fourier transforms on each input feature. Kernels having coefficients of a plurality of trained features are received and on-the-fly Fourier transforms (OTF-FTs) are performed on the coefficients in the kernels. The output of each Fourier transform and each OTF-FT are multiplied together to generate a plurality of products and each of the products are added to produce one sum for each output feature. Two-dimensional inverse Fourier transforms are performed on each sum.

BACKGROUND

Deep learning techniques such as convolutional neural networks (CNNs)are becoming popular for image classification, pixel labeling, pathplanning, and other applications with broad usage in multipleapplications including automotive, industrial, drones, and roboticapplications. Typical CNN architecture includes a cascade of multiplelayer functions, such as convolutions, non-linearity, spatial pooling,and fully connected layers. Convolution layers consisting oftwo-dimensional convolutions typically constitute more than 90% of thecomputations required for image classification and pixel labeling.

One approach for convolution is based on processing in the frequencydomain (also known as the FFT domain) using fast Fourier transform (FFT)techniques. In this technique, convolution is simplified to pointwisecomplex multiplication in an FFT domain. One of the problems with theFFT-based approach is that it generates a large number of frequencydomain coefficients that is proportional to the dimensions of the imagebeing analyzed rather than the size of a filter applied to the image,assuming the filter dimensions are much smaller than the imagedimensions. The memory required to store the frequency domaincoefficients is very high and beyond the typical size of on-chip memoryof most computers and/or processing chips, making this techniqueimpractical for most hardware.

SUMMARY

A method for analyzing images to generate a plurality of output featuresincludes receiving input features of the image and performing Fouriertransforms on each input feature. Kernels having coefficients of aplurality of trained features are received and on-the-fly Fouriertransforms (OTF-FTs) are performed on the coefficients in the kernels.The output of each Fourier transform and each OTF-FT are multipliedtogether to generate a plurality of products and each of the productsare added to produce one sum for each output feature. Two-dimensionalinverse Fourier transforms are performed on each sum.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an imaging system that implements aconvolutional neural network (CNN).

FIG. 2 is a block diagram of a CNN with two convolution layers and afully connected layer followed by a softmax layer.

FIG. 3 is a block diagram showing two-dimensional convolution of aninput feature vector.

FIG. 4 is a diagram showing an example of image analysis in aconvolutional neural network using a fast Fourier transform (FFT) basedapproach.

FIG. 5 is a flowchart that summarily describes an example of theconvolution methods disclosed herein.

FIG. 6 is a diagram describing an example of the image analysis of FIG.4 in greater detail.

FIG. 7 is a spatial two-dimensional array describing the pruning processand the expansion processes of FIG. 6

FIG. 8 is a data flow diagram showing an on-the-fly Fourier transformyielding imaginary and real components.

FIG. 9 is a chart showing a detailed process for performing theon-the-fly Fourier transform of FIG. 8.

DETAILED DESCRIPTION

Image identification/analysis systems identify and/or classify images byimplementing deep learning techniques such as convolutional neuralnetworks (CNNs). Advances in deep learning techniques, such as CNNs, areavailable in part because of advances in computation power along withlow-cost and powerful cameras. A typical CNN architecture includesmultiple layers of two-dimensional convolutions, which typicallyconstitute more than 90% of the total computations necessary for imageanalysis, such as classification, pixel labeling, and other functions.CNNs and imaging systems are described herein that perform convolutionin the frequency domain using fast Fourier transforms (FFTs) withreduced complexity and reduced memory usage.

FIG. 1 is a block diagram of an imaging system 100 that implements aCNN. The imaging system 100 includes a camera 102 that captures images.The camera 102 outputs digital image data representative of the capturedimages to a CNN 104 that performs image analysis techniques, such asclassification described herein. The disclosure described herein isfocused on performing convolution in a frequency domain (sometimesreferred to as the FFT domain) using FFTs. Accordingly, the CNN 104implemented in the imaging system 100 may perform more tasks thandescribed herein. The CNN 104 generates data or output features based onthe analysis of images received from the camera 102. The data istransferred to an output device 106 (or output-image identificationdevice 106) that provides a notification for given image analysis,wherein the notification may include identification of objects in theimage. For example, the output device 106 may be implemented withsoftware, a voice system, or a display that translates the datagenerated by the CNN 104 to data that is readable by a computer oroutput to a speaker or a display for given image analysis.

FIG. 2 is a block diagram of an example of the CNN 104 of FIG. 1. TheCNN 104 includes two convolution layers, referred to as a firstconvolution layer 200 and a second convolution layer 204. The layersdescribed herein perform functions that may be performed by modules orthe like in a hardware or computer system. Max pooling is performed by amax pooling layer 208 located between the first and second convolutionlayers 200 and 204. The CNN 104 includes a fully connected layer 210wherein all the neurons are interconnected in the fully connected layer210. A softmax layer 214 follows the fully connected layer 210 andprovides a final score or probability for a given object class. Theobject class identifies an object in the image or provides thelikelihood as to the identification of an object in the image. Forexample, the object class may be a car or a cat and the softmax layer214 may provide a probability that the object in the image is actually acar or a cat. The CNN 104 described in FIG. 1 is an example of one of aplurality of CNNs that may implement the improvements described herein.CNNs implemented in practice may vary in many ways including the numberof layers, the type of layers, the breadth of each layer, and the orderof layers. The methods and apparatuses described herein are applicablefor variations of CNNs in computations of two-dimensional convolution.

The CNN 104 receives input features 220 (sometimes referred to as inputfeature vectors) that are representative of an image that is to beanalyzed. The features 220 may include pixel values of an imagegenerated by the camera 102, FIG. 1. The features 220 in the example ofFIG. 2 are pixel values in individual color domains. Accordingly, inmany examples, the features 220 have three layers or planes representingthe red, green, and blue color domains. The features 220 are representedas having dimensions I_(X), I_(Y), and I_(Z), wherein I_(Z) is equal tothe three color domains. The features 220 are convolved at the firstconvolution layer 200 with a set of pre-trained receptive filtercoefficients 224 that are also know as field kernels. The coefficients224 are obtained during training processes a priori (offline) andrepresent knowledge of objects of interest used for analysis as is knownin the art. The convolution layer 200 may be followed by a non-linearoperation layer (not shown) that is typically a rectified linear unit(ReLU) or similar function.

Max pooling is applied to the results of the first convolution layer 200at the max pooling layer 208. The max pooling layer 208 introducesslight translation invariance and reduces the feature vector sizeresulting from the first convolution layer 200. In some examples, themax pooling layer 208 filters certain blocks in the input by selectingthe maximum values in the blocks and discarding the remaining values.The max pooling layer 208 may operate independently on every layer ofthe input and resizes the input. In some examples, a pooling layer withfilters having sizes of two by two is applied with a stride of twodownsamples at every depth slice in the input by two along both widthand height, discarding 75% of the values resulting from the convolutionlayer 200. Some examples of the CNN 104 do not include the max poolinglayer 208.

A second convolution is performed by the second convolution layer 204after the max pooling layer 208. The result of the second convolution issometimes referred to as a learned feature vector or an output feature230. The learned feature vector 230 is input to the fully connectedlayer 210 for classification. The softmax layer 214 normalizes theresults generated by the fully connected layer 210. For example, thesoftmax layer 214 maps the vectors to a probability of a given output,which in the example of FIG. 2 includes classification or identificationof an image. In the example described above, the softmax layer 214 mayprovide a probability that the image received from the camera 102, FIG.1, is a car and a probability that the image is a cat. There aremultiple network topologies, such as LeNet5[3], AlexNet[4], Sermanet[5],that have multiple convolution steps and fully connected stages whichresult in large computational complexity. The devices and methodsdescribed herein overcome or reduce the complexity issues.

Two-dimensional convolution in the first convolution layer 200 and thesecond convolution layer 204 uses tremendous resources, which thedevices and methods described herein reduce or alleviate.Two-dimensional convolution is defined by equation (1) as follows:

Output(m,n)=Σ_(i)Σ_(j)(Input(m−i,n−j)×weight(i,j))  Equation (1)

where m, n, i, and j are indices of values in a given array; the indicesi,j have values in the range of 0 to the filter tap, which is also knownas the training parameters, kernel, filter coefficients 224, and otherterms; the indices m,n have values in the range of 0 to the imageheight/width; the “input” is the input features; and the “weight” is thefilter coefficients obtained during training.

FIG. 3 is a block diagram showing two-dimensional convolution of anexample of one of the input features 220 of FIG. 2. The input feature220 has a size (horizontal×vertical) of 16×16 and can vary based onimage dimension (e.g. 1920×1080). The input feature 220 is convolvedwith a weighted kernel 300 that has a plurality of weighted coefficients302. In the examples of FIG. 3, the weighted kernel 300 has a size of5×5, meaning it has twenty-five weighted coefficients 302 and can varybased on the CNN (e.g. 3×3, 7×7, 9×9, etc.). The convolution produces anoutput feature vector 304 having a size of 16×16, which assumes zeropadding along the border. In direct convolution, the two-dimensionalconvolution is implemented per equation (1). In direct convolution, anefficient solution takes account of two-dimensional spatial locality forfetching data from external memory to internal memory. In CNNarchitectures, there are a large number of two-dimensional convolutionsfor each layer, which is equal to the number of input features (N_(i))multiplied by the number of output features (N_(o)). It is not uncommonfor a single convolution operation to be required to store 100 Mbytes offilter coefficients for these convolutions, which overwhelms mostcomputational systems.

FIG. 4 is a diagram 400 showing convolution in neural networks using aFourier transform (FT) based approach, which is described herein beingimplemented as a fast Fourier transform (FFT) based approach. The FFTapproach reduces the complexity of convolution when categorizing imagesin terms of multiplication and addition operations. As per signalprocessing theory, a convolution operation in the time domain becomes apointwise complex multiplication operation in the frequency domain,which is referred to herein as the FT domain or the FFT domain. In theFFT approach, input features 220 are received and have two-dimensionalFFTs 404 applied to them, which yields two-dimensional complex arrays(not shown in FIG. 4) of identical dimension (height/width) as the inputfeatures 220. Weighted kernel coefficients 406 (also known as trainedfeatures) have a two-dimensional FFT 410 performed on them, which alsoyields two-dimensional arrays of complex coefficients having the samedimensions as the input features 220 rather than the dimensions of thekernel coefficients 406. This is to provide for pointwisemultiplication, where the dimensions of both the input features 220 andthe kernel coefficients 406 have to be identical. Assuming thedimensions of the kernel coefficients 406 are typically small comparedto the dimension of the input features 220, the kernel coefficients 406are zero padded to make the dimension of the input features 220 equal tothe dimensions of the kernel coefficients 406. In contrast, when theinput features 220 are smaller than the kernel coefficients 406, thezero padding is applied to the input features 220, so the dimensions ofthe input features 220 match the dimensions of the kernel coefficients406. Typically, the FFT 410 is performed on the weighted kernelcoefficients 406 offline during training. The coefficients need to bestored, which again creates a tremendous number of coefficients andoverwhelms the memory of most computational systems.

The results generated by the FFTs 404 and 410 are multiplied together byway of complex multiplication 414. All the multiplication results areadded together by way of complex addition 416. The complex sumsgenerated by the complex addition 416 have inverse Fourier transforms(IFTs) applied to them, which are implemented in the example of FIG. 4as inverse fast Fourier transforms (IFFTs) 420. The results of the IFFTs420 are the output features 230, which become input features for thenext layer. In situations where the 2D-IFFTs 420 are performed prior tothe operations of the fully connected layer 210, FIG. 2, the results ofthe IFFTs 420 are referred to as learned feature vectors. The primaryadvantage of the FFT approach applied to CNNs is that the 2D-FFTs 404are computed once per input feature, while IFFTs 420 need to be computedonly once per output feature, amortizing the cost of the 2D-FFTs 404 andthe IFFTs 420. In some examples, the FFT approach described hereinprovides a complexity reduction of up to 40% depending on the CNNnetwork structure.

The FFT based approach for convolution described in FIG. 4 isimpractical in many situations due to growth in the memory size requiredto store the weighted kernel coefficients 406. The methods and devicesdescribed herein provide methods for performing convolution in the FFTdomain that reduce the size of the weighted kernel coefficients 406, sothey are identical to the space domain for direct convolution. Themethods described herein include performing on-the-fly Fouriertransforms (OTF-FTs) on the weighted kernel coefficients 406 to preventexplosive growth in the size of storage. More specifically, the OTF-FTsenable frequency domain or FFT domain storage, which is much less thanthe storage required in the case of offline FFTs. Some examples of themethods described herein perform FFTs of the input features 220, pruningof the outputs of the FFTs 404, pointwise multiplication with the outputof the OTF-FT, and expanding the results of the complex addition 416.The methods disclosed herein operate at a rate that is typically four toeight times faster than traditional FFT based approaches. The methodsdescribed herein may be applied to time/space domain analysis inapplications other than two-dimensional image processing. In someexamples, the devices and methods described herein are applicable tosequences of images, such as video.

FIG. 5 is a flowchart 500 that summarily describes examples of theconvolution methods performed in the FFT domain disclosed herein. Instep 502, the two-dimensional FFT is performed on an input feature. Thismay be similar or identical to the 2D-FFTs 404 of FIG. 4. In step 504,the results of the two-dimensional FFTs are pruned or reduced asdescribed in greater detail below. In step 506, the weighted kernelcoefficients 406 are received from an offline training procedure. AnOTF-FT is performed on the coefficients at step 510. In step 512,pointwise multiplication is performed between the pruned results fromstep 504 and the results from the OTF-FT from step 510. The results ofthe pointwise multiplication of step 512 are added by way of complexaddition in step 513. The results of the complex addition are expandedin step 514 as described in greater detail below. In step 516, atwo-dimensional inverse FFT is performed on the results of step 514 toyield the output features or the learned feature vectors.

FIG. 6 is a chart 600 describing an example of the convolution method ingreater detail. The example of FIG. 6 has N_(i) input features 220,which are shown as input feature 1, input feature 2, and input featureN_(i). The example of FIG. 6 has N_(o) output features 230, which areshown as output feature 1, output feature 2, and output feature N_(o). Aplurality of weighted kernel coefficients 406 are also received asinputs from offline training as is known in the art. The kernelcoefficients 406 are sometimes referred to as coefficients of trainedfeatures. The number of kernel coefficients 406 is equal to the productof N_(i) and N_(o). The kernel coefficients 406 are referred to as beingin a spatial domain as there are a plurality of two-dimensional arraysof kernel coefficients 406. The chart 600 is described as performingprocesses, which may be implemented as modules in a computer system.

In the example of FIG. 6, the input features 220 are two-dimensionalimages or color domains of two-dimensional images. In other examples,the input features 220 are output features 230 from a previous layer.Each of the input features 220 has a 2D-FT applied to it. In theexamples described herein the 2D-FTs are implemented as 2D-FFTs 604. Theresults of each of the 2D-FFTs 604 are symmetric, so they can be prunedin a pruning processes 608 as described in greater detail below. Theresults of the pruning processes 608 are input to complex multiplicationprocesses 610. The kernel coefficients 406 have OTF-FTs 614 applied tothem. The OTF-FTs 614 alleviates the need to store a vast amount ofcoefficients such as when the kernel coefficients 406 had FTs applied tothem in an earlier process. The storage requirements for the kernelcoefficients 406 are identical to spatial domain direct convolutionstorage requirements because spatial domain kernel coefficients arebeing stored. The results of the OTF-FTs 614 are input to the complexmultiplication processes 610 where pointwise complex multiplication isperformed on the results of the OTF-FTs 614 and the results of thepruning processes 608. Each of the input features 220 has the processesdescribed herein applied for their contribution to each of the outputfeatures 230, resulting in the product of N_(i) and N_(o) convolutions.Each convolution is a pointwise complex multiplication having dimensionsof the input feature (assuming the dimensions of the input features 220are larger than the dimensions of the kernel coefficients 406 and viceversa) with the output from the OTF-FT 614 of the kernel coefficients406.

The results of the complex multiplication processes 610 are addedtogether by complex addition processes 620. There are N_(i) complexaddition processes 620, one for each of the output features 230. Theresults of the complex addition processes 620 are expanded as describedbelow by expansion processes 622. In summary, the expansion processes622 are performed based on the symmetric nature of the FFT. Onceexpanded, each feature has a two-dimensional inverse Fourier transformapplied, which is implemented as two-dimensional inverse fast Fouriertransforms (2D-IFFT) applied by 2D-IFFT processes 628. The convolutionmethod of FIG. 6 achieves convolution in the frequency or FFT domainwith spatial domain storage of weighted kernel coefficients 406 andavoids large size of frequency domain storage weighted kernelcoefficients 406 as with conventional FFT processes.

FIG. 7 is a spatial two-dimensional array 700 describing the pruningprocess 608 and the expansion processes 622 of FIG. 6. The array 700shows the two-dimensional (height H and width W) results of the Fouriertransform (the 2D-FFTs 604 or the OTF-FTs 614). The results extend in avertical direction for the height H and a horizontal direction for thewidth W, wherein each box represents a complex number resulting from theFourier transform. A point 702 is the subject of the followingdescription. The shaded portions 710 and 712 are identical due tosymmetry in the Fourier transforms with arrows 720 representing columnshaving matching values. Lettered blocks are examples of matching complexnumbers. For example, the complex numbers represented by the letters A,B, C, and D are representative of the locations of matching complexnumbers.

The pruning process 608 deletes the coefficients in the shaded portion712. For example, for the array 700 with input feature dimensions ofheight of 256 and width of 256, the pruning process yields an arrayhaving a width W of 129 (half the original width plus one) in the widthdirection and 256 in height H. Accordingly, the number of coefficientsmultiplied by the complex multiplication processes 610 originating fromthe input features 220 with the results of the OTF-FTs 614 are reducedto almost half of their original sizes.

FIG. 8 is a data flow diagram 800 showing the input and output for theOTF-FT 614 of FIG. 6. A two-dimensional array of coefficients 802 havinga height H and a width W has a spatial domain kernel 804 input having aheight H-K and a width W-K and zero padding to match size (height andwidth) of the input feature. Due to symmetry and pruning processdescribed above, the OTF-FT 614, FIG. 6, is processed with a reducedwidth of W/2+1 as described above. This means the OTF-FT 614 needs to becalculated for H×(W/2+1), which reduces the processing and storagerequirements by almost half. The results of the OTF-FT 614 yield complexnumbers, which are two sets of coefficients, a real set of coefficients810 and an imaginary set of coefficients 812. Both sets of coefficients810 and 812 are arranged in arrays that are H high by W/2+1 wide.

FIG. 9 is a diagram 900 showing the OTF-FT process in detail. In a firststep, the pruned array 802 of kernel coefficients is split so only thetop half is processed. In other examples, only the bottom half isprocessed. The splitting forms a split array 904. The split array 904,which is the top half, is processed by a one-dimensional horizontaldirect Fourier transform (1D-HDFT) 908. The 1D-HDFT 908 takes advantageof zero weight and real values in the split array 904 for efficiency. DCfrequency calculations are removed, which reduces the number ofmultiplications required in the 1D-HDFT 908. The results of the 1D-HDFT908 are split into a real component 912 and an imaginary component 914.

The real component 912 is processed by a one-dimensional vertical directFourier transform (1D-VDFT) 920 to yield results in a first array 922 ofcomplex numbers. The imaginary component 914 is processed by a 1D-VDFT924 to yield results in a second array 926 of complex numbers. The1D-VDFTs 920 and 924 rely on zero weight in the kernel andsimplification of the DC frequency calculations to reduce the number ofmultiplication steps required to generate the results in the first andsecond arrays 922 and 926. A merge operation 930 merges the first array922 and the second array 926 using complex addition of both arrays andextends both planes in the vertical direction due to the symmetricproperty to yield the final results. The final results are a real plane934 and an imaginary plane 936 each having dimensions of the arrays 810and 812 of FIG. 8. The arrays are processed per FIG. 6.

The convolution and other operations described herein may be performedin software, hardware, firmware processors, microprocessors, ASICs,DSPs, computer-readable media, etc or any other means or devices asknown by those skilled in the art. The operations described hereinimprove image identification and other processes that involveconvolution. When applied to computer programs, the processes describedherein significantly improve the operation of the computer by enablingconvolution and image analysis in the FFT domain without the creation ofan excessive number of coefficients that overwhelms the computer. It isnoted that some of the processes have been described herein as usingFFTs, these processes may be performed using standard FourierTransforms. The layers described herein perform functions that may beperformed by modules or the like in a hardware or computer system.

Although illustrative embodiments have been shown and described by wayof example, a wide range of alternative embodiments is possible withinthe scope of the foregoing disclosure.

What is claimed is:
 1. A system for analyzing images, the devicecomprising: an input for receiving input features of an image; Fouriertransform (FT) modules, each FT module for performing a two-dimensionalFT on each input feature of the image; one on-the-fly Fourier transform(OTF-FT) module for each input feature, each OTF-FT module forperforming Fourier transforms on weighted coefficients of a plurality oftrained image features; complex multiplication modules for performingcomplex multiplication on the output of each FT and each OTF-FT; complexaddition modules for performing complex addition on each output of eachof the complex multiplication modules; and two-dimensional inverseFourier transforms (2D-IFTs) for performing two-dimensional inverseFourier transforms on the outputs of the complex addition.
 2. The systemof claim 1, further comprising pruning modules for reducing the numberof coefficients generated by each of the FT modules.
 3. The system ofclaim 2, further comprising expansion modules for expanding thecoefficients generated by the complex addition modules to correspond tothe number of coefficients reduced by the pruning modules.
 4. The systemof claim 1, wherein each feature of the image is a color plane of theimage.
 5. The system of claim 1, wherein each of the OTF-FT modulescomprises: a module for performing a one-dimensional horizontal directFourier transform (1D-HDFT); and a module for performing aone-dimensional vertical direct Fourier transform (1D-VDFT) on theresult of the 1D-HDFT.
 6. The system of claim 1, wherein the OTF-FTscomprise: a module for performing a one-dimensional horizontal directFourier transform (1D-HDFT) on one half of the coefficients in thekernel; a module for splitting the results of the 1D-HDFT into a realcomponent and an imaginary component; a module for performing aone-dimensional vertical direct Fourier transform (ID-VDFT) on the realcomponent; a module for performing a one-dimensional vertical directFourier transform (ID-VDFT) on the imaginary component; and a module formerging the real component and the imaginary component.
 7. The system ofclaim 6, further comprising a module for expanding the results of themerging to generate an array equal in size to the kernel.
 8. The systemof claim 1, wherein the multiplication module is for performingpointwise multiplication.
 9. The system of claim 1, further comprising amodule for pruning the results of the FTs.
 10. A method for analyzingimages to produce a plurality of output features, the method comprising:receiving a plurality of input features related to the image; performingFourier transforms (FTs) on each of the plurality of input features;receiving a plurality of kernels, each having coefficients for trainedfeatures; performing one on-the-fly Fourier transform (OTF-FTs) on thekernel coefficients of each kernel; multiplying the output of each FTand each OTF-FT together to generate a plurality of products; addingeach of the products to produce one sum for each output feature; andperforming two-dimensional inverse Fourier transforms (2D-IFTs) on eachsum to generate each output.
 11. The method of claim 10, wherein theinput features are color planes of the image.
 12. The method of claim10, further comprising pruning the results of the FTs and expanding theresults of the adding.
 13. The method of claim 10, wherein performingthe OTF-FTs comprises: performing a one-dimensional horizontal directFourier transform (1D-HDFT); and performing a one-dimensional verticaldirect Fourier transform (ID-VDFT) on the result of the 1D-HDFT.
 14. Themethod of claim 10, wherein performing OTF-FTs comprises: performing aone-dimensional horizontal direct Fourier transform (1D-HDFT) on onehalf of the coefficients in the kernel; splitting the results of the1D-HDFT into a real component and an imaginary component; performing aone-dimensional vertical direct Fourier transform (ID-VDFT) on the realcomponent; performing a one-dimensional vertical direct Fouriertransform (ID-VDFT) on the imaginary component; and merging the realcomponent and the imaginary component.
 15. The method of claim 14,further comprising expanding the results of the merging to generate anarray equal in size to the kernel.
 16. The method of claim 14, furthercomprising pruning the kernel prior to performing the 1D-HDFT.
 17. Themethod of claim 10, wherein performing the multiplication comprisesperforming pointwise multiplication.
 18. A method for identifying animage, the method comprising: receiving input features of the image;performing fast Fourier transforms (FFTs) on each feature of the image;pruning the results of the FFT; receiving a plurality of kernels havingcoefficients corresponding to trained image features; performing oneon-the-fly Fourier transform (OTF-FTs) on the coefficients in thekernels; applying pointwise multiplication to the output of each FFT andeach result of the pruning to generate a plurality of products; addingthe products to generate sums; expanding the sums; and performingtwo-dimensional inverse fast Fourier transforms (2D-IFFTs) on each sumto generate a plurality of output features.
 19. The method of claim 18,further comprising analyzing the output features to identify the image.20. The method of claim 19, further comprising analyzing the outputfeatures and assigning at least one probability as to the identity ofthe image in response to the analyzing.